f1[x_, y_] := Sin[x] Sin[y] + y/2 - x

f2[x_, y_] := (y^2) - 4  (x^2) + Cos[x] Cos[y]

J[x_, y_]=(({{(∂_x) f1[x, y], (∂_y) f1[x, y]}, {(∂_x) f2[x, y], (∂_y) f2[x, y]}}));

J[x, y] // MatrixForm

Module[{x, y}, 
x = 1. ; 
      y = 3. ; 
Do[
r = 
          Solve[J[x, y] . {dx, dy} == -{f1[x, y], f2[x, y]}, {dx, 
              dy}]; 
x = 
          x + dx /. r[[1]]; 
y = 
          y + dy /. r[[1]]; 
Print["x = ", 
          N[x, 10], ",  y = ", N[y, 10]], 
{i, 1, 
          4}];
]

N[{Pi/2, Pi}, 8]